Integral of $$$\frac{\cos{\left(u \right)}}{v}$$$ with respect to $$$u$$$

Find $$$\int \frac{\cos{\left(u \right)}}{v}\, du$$$.

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{v}$$$ and $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

$${\color{red}{\int{\frac{\cos{\left(u \right)}}{v} d u}}} = {\color{red}{\frac{\int{\cos{\left(u \right)} d u}}{v}}}$$

The integral of the cosine is $$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$:

$$\frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{v} = \frac{{\color{red}{\sin{\left(u \right)}}}}{v}$$

Therefore,

$$\int{\frac{\cos{\left(u \right)}}{v} d u} = \frac{\sin{\left(u \right)}}{v}$$

Add the constant of integration:

$$\int{\frac{\cos{\left(u \right)}}{v} d u} = \frac{\sin{\left(u \right)}}{v}+C$$

$$$\int \frac{\cos{\left(u \right)}}{v}\, du = \frac{\sin{\left(u \right)}}{v} + C$$$A